On the impact ionization rate in direct gap semiconductors

TL;DR

This paper develops a quantum-mechanical model for impact ionization in direct gap semiconductors, revealing the energy dependence of the ionization rate near the threshold and identifying dominant anisotropic and isotropic contributions.

Contribution

It provides an explicit form of the impact ionization rate near the threshold using a 14-band k·p model, highlighting the cubic term's dominance at room temperature for certain materials.

Findings

Impact ionization rate near threshold is a superposition of anisotropic quadratic and isotropic cubic terms.

The cubic term dominates at room temperature for direct-gap semiconductors with E_g up to 1.5 eV.

Explicit coefficients for the impact ionization rate are derived within the 14-band k·p framework.

Abstract

We present quantum-mechanical theory of impact ionization in semiconductors with the direct band gap in \$\Gamma\$-point. It is shown that energy dependence of the impact ionization rate \$\mathcal{W}(E)\$ near a threshold \$E_{th}\$ is given by superposition of the two terms, one of which is strongly anisotropic and quadratic in \$E-E_{th}\$, while another one is isotropic and cubic in \$E-E_{th}\$. Explicit form of the coefficients in such representation is derived in the framework of the 14-band \${\bf k\cdot p}\$ model, and we claim the room temperature domination of the cubic contribution for most of the direct-gap materials with \$E_g\$ up to 1.5 eV.

Equations

$$\mathcal{W}(E)=C(E-E_{th}{)}^n,$$

$$W_{\{0,3\}\to \{1,2\}}=\frac{2\pi }{\hslash }{\left|⟨{\alpha }_1{\alpha }_2\left|\hat{V}\right|{\alpha }_0{\alpha }_3⟩\right|}^2\delta (\Delta \mathcal{E}),$$

$$\mathcal{W}={\tau }^{-1}=\sum _{{\alpha }_1,{\alpha }_2,{\alpha }_3}{W}_{\{0,3\}\to \{1,2\}}.$$

$$\mathcal{W}={\mathcal{W}}_2+{\mathcal{W}}_3=A(E-E_{th}{)}^2+B(E-E_{th}{)}^3,$$

$$\begin{gathered} \mathcal{W}=\frac{\pi \hslash F\left(\frac{{\Delta }_0}{Eg}\right)}{12m_e{E}_g^2}{\left(\frac{4\pi e^2}{\varkappa }\right)}^2\int \frac{d^3{q}_1{d}^3{q}_2}{(2\pi {)}^6}[{\mathcal{I}}_{cv}({\mathbf{q}}_1,{\mathbf{q}}_3)+ \\ {\mathcal{I}}_{cv}({\mathbf{q}}_2,{\mathbf{q}}_3)]\delta \left(q_1^2+q_2^2-\frac{2m_e(E_0-E_{th})}{{\hslash }^2}\right), \end{gathered}$$

$$F(x)=\frac{(1+x{)}^2(1+x/3{)}^3}{(1+7x/9+x^2/6)(1+2x/3{)}^2(1+x/2)}.$$

$$B=\frac{{\omega }_B^{\ast }}{18E_g^3}\frac{E_g+{\Delta }_0}{E_g+\frac{2}{3}{\Delta }_0}F\left(\frac{{\Delta }_0}{E_g}\right),$$

$$A=\frac{4}{3}\frac{{\omega }_B^{\ast }}{E_G^2}\frac{Q^4}{P^4}{J}_4(\mathbf{u})\frac{E_g+\frac{1}{2}{\Delta }_0}{E_g+\frac{1}{3}{\Delta }_0}F\left(\frac{{\Delta }_0}{E_g}\right),$$

$$\mathcal{R}=\sum _{{\alpha }_0}\mathcal{W}(E({\mathbf{k}}_0)){\tilde{f}}_{{\alpha }_0},$$

$${\mathcal{R}}_i=\mathcal{D}(E_{th})\mathcal{N}{\int }_{E_{th}}^{\infty }\overline{{\mathcal{W}}_i(E)}\exp (-E/\mathcal{T})dE,$$

$${\mathcal{T}}^{\ast }=\frac{\overline{A}}{3B}=\frac{16}{35}\frac{Q^4}{P^4}\frac{E_g^3}{E_G^2}G\left(\frac{{\Delta }_0}{E_g}\right),$$

Full text

, - ( . . 1), , - (IMPATT), (APD) Sze , (I-MOS) Gopalakrishan , 5 mV/dec T=400 K, . $E_{g}\lesssim 1\textrm{ eV}$, : $E_{th}\approx E_{g}(1+2\mu)$ APY , $\mu=m_{e}/m_{hh}\ll 1$. , , . , ’’ ’’ , . $C$, $n$

[TABLE]

$E_{th}$. $\mathcal{W}(E)$ ($n=2$), . .  1959  . Keldysh ,  Choo ,Harrison , $\mathcal{W}(E)$ In0.53Ga0.47As , $n=2.5$ Choo , $n=4.3$ Choo $n=5.6$ Harrison . ,  Gelmont , , , $\rm{Cd}_{0.2}\rm{Hg}_{0.8}\rm{Te}$ $E_{g}\approx 0.1\rm{eV}$, $\mathcal{W}(E)$ , $\mathcal{W}(E)$ , ( , ).

, $\mathcal{W}$ $E-E_{th}$. , , , , . , , $E_{g}\sim 1\,\textrm{eV}$, , .

I .

, :

[TABLE]

. 1, $\hat{V}$ – , $\Delta\mathcal{E}=E_{1}+E_{2}-E_{3}-E_{0}$  footnoteE0 , $\alpha_{i}=\{{\bf k}_{i},\xi_{i}\}$ – : . 0 (  (1))

[TABLE]

, ,  Dmitriev , meV, $\mathcal{W}(E)$ , ,

[TABLE]

, , $A$ $E_{g}$ . , , $E_{g}/\mathcal{T}$, $\mathcal{T}$ – .

Zegrya , ${\bf k}_{i}^{th}$ ${\bf q}_{i}={\bf k}_{i}-{\bf k}_{i}^{th}\ll{\bf k}_{i}^{th}$. $q_{i}$ $\mathcal{W}(E)$

[TABLE]

$\mathcal{I}_{cv}({\bf q}_{i},{\bf q}_{j})=I_{cv}({\bf k}_{i}({\bf q}_{i}),{\bf k}_{j}({\bf q}_{j}))$, $I_{cv}$ , ${\bf q}_{3}={\bf q}_{1}+{\bf q}_{2}-{\bf q}_{0}$ , $q_{0}=\left(\frac{\partial E_{0}}{\partial k_{0}}\right)_{th}^{-1}(E_{0}-E_{th})$ ${\bf k}_{0}^{th}$, $\Delta_{0}$ – - ,

[TABLE]

(5) , $\mathcal{W}(E)$ $E-E_{th}$ $\mathcal{I}_{cv}$ ${\bf q}_{i}$. , $\mathcal{I}_{cv}({\bf 0},{\bf 0})=I_{cv}({\bf k}_{i}^{th},{\bf k}_{j}^{th})$,  Anderson , $\mu=m_{e}/m_{hh}\ll 1$  APY . , ( ) . $A$ $B$  (4) $\mathcal{I}_{cv}$, (${\bf q}_{i}$). 14- ${\bf k}\cdot{\bf p}$  Winkler , , eV $E_{c}$ ( . . 2). ${\bf k}\cdot{\bf p}$ (c) (v), , . . ,  Kane , Greshnov . $B$

[TABLE]

$\Delta_{0}\gg E_{g}$  Gelmont ( $\omega_{B}^{*}=m^{*}e^{4}/2\varkappa^{2}\hbar^{3}$ – ). , , 14- ${\bf k}\cdot{\bf p}$ .

$E_{g},\Delta_{0},\Delta_{G}$ (v) (c*′*), $\mathcal{I}_{cv}$ :

[TABLE]

P Q – c-v c*′-v ( . 2), $J_{4}({\bf u})=I_{1}({\bf u})-3I_{1}^{2}({\bf u})$ – $I_{1}({\bf u})=u_{x}^{2}u_{y}^{2}+u_{x}^{2}u_{z}^{2}+u_{y}^{2}u_{z}^{2}$, ${\bf u}={\bf k}_{0}/k_{0}$ – . , 14- ${\bf k}\cdot{\bf p}$- (c′*), . , . 3. , , [100] [111]. , $A$ $E_{G}$ $B$, , , .

II .

[TABLE]

• , , ${\tilde{f}}_{\alpha}=f_{\alpha}-f_{0}(E({\bf k}))$. ( , )  (9) $\mathcal{R}$, $\mathcal{R}_{2,3}$, (4) ${\tilde{f}}(E)=\mathcal{N}\exp(-E/\mathcal{T})$. $\mathcal{T}$ ( ). $\mathcal{R}_{2}$ $\mathcal{R}_{3}$

[TABLE]

, $\mathcal{D}(E)$ – . $\overline{J_{4}({\bf u})}=2/35$ $\mathcal{R}_{2,3}$,

[TABLE]

$G(x)=\frac{(1+2x/3)(1+x/2)}{(1+x)(1+x/3)}$. InAs Winkler , $0.4\,\textrm{eV}$, ${\mathcal{T}}^{*}\approx 10\,\textrm{K}$, 50 . (11), ${\mathcal{T}}^{*}=300\,\textrm{K}$ $E_{g}\lesssim 1-1.5\,\textrm{eV}$, .

, $E_{g}$ $E_{G}$ $\Delta_{0}/E_{g}$, . , - - $E_{so}({\bf k})$ $\Delta_{0}\rightarrow 0$ , . , . (8) $\Delta_{0}/E_{g}\gg E_{g}/E_{G}$, 1 eV, GaAs $\beta=\Delta_{0}E_{G}/E_{g}^{2}$ 0.8, InP – 0.25. , , $\overline{A}$ (8) 10 %, . , 300 K. , , $\beta\ll 1$, - .

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